A conditional probability approach to M/G/1-like queues

Transcription

1 Performance Evaluaton 65 (2008) A condtonal probablty approach to M/G/1-lke queues Alexandre Brandwan a,, Hongyun Wang b a Department of Computer Engneerng, Baskn School of Engneerng, Unversty of Calforna, Santa Cruz, CA 95064, USA b Department of Appled Mathematcs & Statstcs, Baskn School of Engneerng, Unversty of Calforna, Santa Cruz, CA 95064, USA Receved 29 August 2006; receved n revsed form 5 May 2007; accepted 20 September 2007 Avalable onlne 5 October 2007 Abstract Followng up on a recently renewed nterest n computatonal methods for M/G/1-type processes, ths paper consders an M/G/1-lke system n whch the servce tme dstrbuton s represented by a Coxan seres of memoryless stages. We present a novel approach to the soluton of such systems. Our method s based on condtonal probabltes, and provdes a smple, computatonally effcent and stable approach to the evaluaton of the steady-state queue length dstrbuton. We provde a proof of the numercal stablty of our method. Wthout explct use of matrx-geometrc technques or stochastc complementaton, we are able to handle systems wth state-dependent servce and arrval rates. The proposed approach can be used to compute the queue length dstrbuton for both fnte and nfnte M/G/1-lke queues. In the case of an nfnte, state-ndependent queue, our method allows us to show usng elementary tools that the queue length dstrbuton s asymptotcally geometrc. The parameter of the asymptotc geometrc can be expressed through a smple set of equatons, easly solved usng fxed pont teraton. Our approach s very thrfty n terms of memory requrements, easy to mplement, and generally fast. Numercal examples llustrate the performance of the proposed method. c 2007 Elsever B.V. All rghts reserved. Keywords: M/G/1-lke queues; Coxan dstrbuton; State-dependent servce; Condtonal probablty; Queue length dstrbuton; Recurrent soluton; Numercal stablty; Computatonal effcency; Asymptotc geometrc dstrbuton; Algorthms; Performance 1. Introducton M/G/1-type queues are an essental tool n the performance analyss of computers and computer networks. Devces rangng from volumes n a storage subsystem to statons n an optcal rng network have been represented as M/G/1 queues. Oftentmes, t s necessary to take nto account the fnte buffer space at such devces (e.g. to properly sze the buffers), and, n some systems, the fact that the servce rate may depend on the number of customers n the system (e.g. due to contenton for shared resources). The effcent and numercally stable soluton of such queues, n partcular, the calculaton of the statonary queue length probabltes, s therefore of mportance. Recently, there has been a renewed nterest n computatonal methods for M/G/1-type processes 27,30. As dscussed by Stathopoulos et al. 30, matrx analytc technques ntroduced by Neuts 23,24 are often used to evaluate such processes. In partcular, Ramaswam s algorthm 25,21 provdes a numercally stable approach to the computaton of steady-state probabltes n an M/G/1-type system. Ths algorthm s based on stochastc complementaton 28, and ts fastest mplementaton uses the FFT 22 (cf. 30). The ETAQA method 27 has been proposed as an alternatve for the Correspondng author. Tel.: E-mal addresses: alexb@soe.ucsc.edu (A. Brandwan), hongwang@ams.ucsc.edu (H. Wang) /$ - see front matter c 2007 Elsever B.V. All rghts reserved. do: /.peva

2 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) computaton of the moments of the queue length dstrbuton (but not the dstrbuton tself). Stathopoulos et al. 30 derve a new formulaton for ETAQA and demonstrate ts lnks wth the Ramaswam s method. The large body of prevous work ncludes the work by Gaver et al. 14, who consder fnte queues n randomly changng envronments, and derve a numercal method nvolvng recursve determnaton of certan matrces. Brght and Taylor 10 extend the logarthmc reducton algorthm proposed by Latouche and Ramaswam 18 to leveldependent nfnte queues. They note possble numercal problems when recursvely calculatng matrces nvolved n the soluton. Such problems are also mentoned n Gaver et al. 14. Latouche shows n 17 that Newton s method appled to non-lnear equatons n Markov chans s quadratcally convergent although not very attractve because of ts computatonal complexty. Bn and Men 6 extend the cyclc reducton technques to nfnte block matrces, and later propose an mprovement based on FFT to lower computatonal cost and ncrease numercal stablty 7. Akar and Sohraby 1 propose an nvarant subspace approach wth at least quadratc convergence rates and potentally mproved accuracy due to the avodance of truncaton. Ramaswam and Taylor 26 study the generalzaton of matrx-geometrc statonary dstrbuton to level-dependent quas-brth-and-death processes. More recently, Bean et al. 3 consder quasstatonary dstrbutons for level-dependent processes and ther computaton usng methods akn to the Latouche Ramaswam algorthm. Ye 31 studes the theoretcal propertes of the Latouche Ramaswam logarthmc reducton algorthm, pontng out numercal stablty ssues and offerng an alternatve based on a more stable algorthm for nvertng a dagonally domnant matrx. The asymptotc behavor of an nfnte queue has been studed, among others, by Bean et al. 4 who consder quas-brth-and-death processes wth decomposable spaces, and use the matrx-geometrc form of the steady state probablty vector to derve a numerc method for computng the caudal characterstc factor of the process usng matrces smaller than earler tradtonal methods. Knessl et al. 15 consder a state-dependent M/G/1 where the nterarrval and the servce tme dstrbutons depend on the amount of unfnshed work n the system. They apply perturbaton methods to derve approxmatons for several measures pertanng to the unfnshed work and the mean busy perod n such a queue. In a related work, Knessl et al. 16 study a GI/G/1 queue wth a smlar form of state dependence usng the same perturbaton methods. The reader s referred to the books by Latouche and Ramaswam 19, and by Bn et al. 5 for an overvew of propertes of matrx analytc approaches and numercal methods for quas-brth-and-death problems and M/G/1-type Markov chans. Practcal consderatons and software mplementaton for several of the methods mentoned above are dscussed by Bn et al. 8,9. In ths paper, we consder an M/G/1-lke system n whch the servce tme dstrbuton s represented by a Coxan 12 seres of memoryless stages. It s well known that a Coxan dstrbuton can approxmate arbtrarly closely any dstrbuton, and a two-stage Coxan can be used to match the frst two moments of any dstrbuton 2 whose coeffcent of varaton s greater than 1. Several authors have consdered algorthms for matchng an arbtrary dstrbuton by a Coxan, e.g. 11,29,13,20. Our method s based on condtonal probabltes, and provdes a smple, computatonally effcent and numercally stable approach to the evaluaton of the steady-state queue length dstrbuton. Wthout any explct use of matrxgeometrc technques or stochastc complementaton, we are able to solve systems wth state-dependent servce and arrval rates. As a result, the proposed approach can be used for fnte M/G/1-lke queues as well. In the case of an nfnte, state-ndependent queue, the form of the queue length dstrbuton ndcates that t s asymptotcally geometrc wth a coeffcent gven as a soluton of a smple set of equatons, easly solved va fxed pont teraton. To the best of our knowledge, the proposed method s novel. Ths paper s organzed as follows. In Secton 2 we descrbe n more detal the queue consdered and present our computatonal approach. Secton 3 s devoted specfcally to the case of an nfnte state-ndependent queue and ts asymptotcally geometrc queue length dstrbuton. In Secton 4 we outlne the proof of numercal stablty of our method. Secton 5 presents numercal results to llustrate the behavor of our method, and Secton 6 concludes ths paper. Addtonally, n the Appendx we show that the condtonal probablty on whch we base our approach converges to a sngle fxed pont n the case of an nfnte state-ndependent queue. 2. M/G/1-lke model and ts recurrent soluton 2.1. Model consdered and ts equatons The sngle-server queung model consdered s shown n Fg. 1.

3 368 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Fg. 1. M/G/1-type queue wth Coxan-lke servce. We assume that customers arrve from a quas-posson source wth rate λ where n denotes the current number of customers n the system. The servce of a customer s represented by a state-dependent Coxan-lke dstrbuton 12 wth a total of k stages. Both the completon rates of the memoryless stages and the ext probabltes n ths dstrbuton may depend on the number of customers n the queue. We denote by µ the servce rate of stage and by q the probablty that the customer completes ts servce followng stage, = 1,..., k. We let ˆq = 1 q denote the probablty that the customer proceeds to stage + 1 upon completon of stage when there are n customers n the system. We assume that µ > 0, = 1,..., k, 0 < ˆq 1 for < k and ˆq k = 0. Note that our formulaton allows not only the mean value but also the nature of the servce tme dstrbuton to change as the number of customers n the queue changes. We consder the system of Fg. 1 n steady state, and we use the couple (n, ) where n (n 1) s the current number of customers and ( = 1,..., k) s the current stage of servce to descrbe the state of the system when t s not dle. We denote by p(n, ) the correspondng steady state probablty, and by p(0) the probablty that there are no customers n the system. It s straghtforward to obtan the balance equatons for the correspondng state probabltes p(1, 1)λ + µ 1 = p(n + 1, )µ (n + 1)q (n + 1) + p(0)λ(0) (1) =1 p(1, )λ + µ = p(n, 1)µ 1 ˆq 1, > 1 (2) for n = 1, and p(n, 1)λ + µ 1 = p(n + 1, )µ (n + 1)q (n + 1) + p(n 1, 1)λ(n 1) (3) =1 p(n, )λ + µ = p(n, 1)µ 1 ˆq 1 + p(n 1, )λ(n 1), > 1 (4) for n > 1. These balance equatons can be transformed nto equatons for the condtonal probabltes of the servce stage number gven the current number of customers, and solved as a smple recurrence Recurrent soluton Our goal s to transform the standard balance equatons (1) (4) nto easy-to-evaluate recurrence relatons. We denote by p( n) the condtonal probablty that the current servce stage s gven that the number of customers s n, and by p the margnal probablty that the number of customers n our M/G/1-lke queue s equal to n. For all n 1 and = 1,..., k we have p(n, ) = p( n)p. (5) From (5) and the balance equatons (1) (4), t s easy to see that p can be expressed as p = 1 G n λ(l 1)/u(l), l=1 where G s a normalzng constant and u = p( n)µ q =1 (6) (7)

4 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) u s the condtonal rate of completon gven n, and 1/u can be vewed as the mean servce tme gven that there are n customers n the system. Substtutng (5) and (6) nto the balance equatons, and usng the fact that p > 0 for all feasble values of n, we readly obtan the equatons for the condtonal probabltes p( n) p(1 1)λ(1) + µ 1 (1) = λ(1) + u(1) p( 1)λ(1) + µ (1) = p( 1 1)µ 1 (1) ˆq 1 (1), > 1 (9) for n = 1, and p(1 n)λ + µ 1 = λ + p(1 n 1)u (10) p( n)λ + µ = p( 1 n)µ 1 ˆq 1 + p( n 1)u, > 1 (11) for n > 1. Snce we must have p( n) = 1 for n 1, =1 we can express the soluton of (8) and (9) as p( n = 1) = 1 H µ 1 (1) ˆq 1 (1)/λ(1) + µ (1), (13) =2 where H s a normalzng constant. The empty product s equal to 1 by conventon, so that we have p(1 1) = 1/H. Note that Eqs. (8) and (9), together wth the normalzng condton (12), mply that we must have u(1) < µ 1 (1) n the steady state of our system. We now consder Eqs. (10) and (11) for ncreasng values of n = 2, 3,... For each n, the p( n 1) are known from the precedng step, so that (10) and (11) s a system of k equatons for the k condtonal probabltes p( n), = 1,..., k. We note that Eqs. (10) and (11) nvolve u, a lnear combnaton of the condtonal probabltes p( n), and we can express the latter as p( n) = b λ + c u. From (10) we have b 1 = 1/λ + µ 1, and c 1 = p(1 n 1)/λ + µ 1. (15) Eq. (11) yelds the followng recurrence b = b 1 µ 1 ˆq 1 /λ + µ, c = c 1 µ 1 ˆq 1 + p( n 1)/λ + µ for > 1. Havng computed the coeffcents b and c from the above recurrence, we determne u from the normalzng condton (12) / u = 1 λ b c. (17) =1 =1 For n = 1, we can use a smlar approach or the soluton form gven by (13). Note that the recurrence tself (15) and (16) nvolves no subtractons to detract from numercal stablty. A formal proof of stablty s outlned n Secton 4. Space requrements of our recurrent computaton are very modest snce we need only one set (for a sngle value of n) of the condtonal probabltes p( n) at any tme. Even the values of u need not be kept f we embed our recurrence nto the computaton of the margnal probablty p. The latter s gven by the famlar product-form formula (6) dentcal to that of a state-dependent M/M/1 queue wth arrval rate λ and servce rate u. Clearly, f λ vanshes for some value of n, the customer populaton s lmted and the recurrence has a natural stoppng (8) (12) (14) (16)

5 370 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) pont. If the populaton n the queue s unrestrcted, as would be the case wth a Posson source, the computaton wll have to stop when values of p( n) (and thus u) for consecutve values of n no longer change more than some reasonable tolerance we fx as the convergence crteron. To the best of our knowledge, the proposed approach to the computaton of the statonary queue length probabltes n an M/G/1-lke queue s novel. The next secton s devoted to the partcular case of an nfnte state-ndependent queue. 3. Infnte state-ndependent queue We now restrct our attenton to the partcular case of an M/G/1-lke queue wth Posson arrvals and standard Coxan servce dstrbuton. Specfcally, we have λ = λ, µ = µ, and q = q. Snce the approach presented n the precedng secton nvolves a recurrent computaton of the condtonal probabltes p( n) for consecutve values of n, t mght seem that, for the open queue, the method would requre the soluton of an nfnte number of such recurrences. In ths secton we argue that, n practce, these condtonal probabltes tend to quckly reach a lmtng value, so that only a relatvely small number of recurrences need to be solved. In the process, we demonstrate the asymptotcally geometrc form of the steady-state dstrbuton p for large n. For the queue consdered, the equatons for the condtonal probablty for n = 1 become p(1 n = 1)λ + µ 1 = λ + u(1) p( n = 1)λ + µ = p( 1 n = 1)µ 1 ˆq 1, > 1, (19) and for n > 1 we have p(1 n)λ + µ 1 = λ + p(1 n 1)u (20) p( n)λ + µ = p( 1 n)µ 1 ˆq 1 + p( n 1)u, > 1. (21) As n tends to nfnty, the condtonal probablty p( n) and u tend to ther lmtng values whch we denote by p() and ũ, respectvely. A rgorous proof of ths convergence s presented n the Appendx. Gven the form of p n (6), the convergence to lmtng values means that we have asymptotcally for large n, (18) p(n + 1)/p λ/ũ, (22).e., the steady-state dstrbuton s asymptotcally geometrc. In our numercal experments, we found that, typcally, the soluton of Eqs. (20) and (21) tends rapdly to the lmtng dstrbuton p() as n ncreases. Consequently, we can use the followng computatonal approach to obtan the queue length dstrbuton for the nfnte state-ndependent queue. Select a convergence crteron, e.g. p( n) p( n 1) < ε wth some reasonable value of ε, and let ñ be the value of n for whch the convergence s attaned. We can express the steady-state dstrbuton p as n λ/u(l), n ñ p 1 l=1 G ñ λ/u(l) (λ/ũ) n ñ, n > ñ. l=1 The normalzng constant G can be wrtten as ñ 1 n ñ 1 G = 1 + λ/u(l) + λ/u(l) 1 (λ/ũ). n=1 l=1 l=1 Thus, we use the recurrence descrbed n Secton 2.2 (13) (17) for n = 1,..., ñ (.e. smply untl convergence has been reached n the computaton of p( n) and u). Once convergence has been attaned, we use the last computed value for u as the lmtng value ũ n the geometrc tal of the dstrbuton p. Wth ths approach, the expected (23)

6 number of customers n the system can be wrtten as { n 1 ñ ñ np + G 1 (λ/ũ) + (λ/ũ) 1 (λ/ũ) n=1 2 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) ñ λ/u(l) l=1 Clearly, the system s stable as long as λ < ũ. In practce, t has been our experence that convergence to ũ tends occur qute rapdly, resultng typcally n moderate values for ñ. In Secton 5, we present numercal results to llustrate the behavor of our method. Our recurrent algorthms produce the set of u for n = 1,..., ñ,.e., untl convergence to ũ. If desred, the lmtng value ũ can also be computed ndependently as follows. The lmtng condtonal probabltes p() and the lmtng servce rate value ũ satsfy the followng equatons p(1)λ + µ 1 = λ + p(1)ũ p()λ + µ = p( 1)µ 1 ˆq 1 + p()ũ, > 1. (25) Eqs. (24) and (25) can be easly (and, generally, rapdly) solved usng a fxed-pont teraton. Startng wth an ntal probablty dstrbuton p 0 () (we use a superscrpt to denote the teraton number) and the correspondng value ũ 0, we compute values at teraton p (1) λ + p 1 (1)ũ 1 /λ + µ 1 (26) p () p ( 1)µ 1 ˆq 1 + p 1 ()ũ 1 /λ + µ, > 1. (27) At each teraton we normalze the values so as to have k =1 p () = 1. Ths allows us to determne the lmtng values p() and ũ. The latter s the servce rate value for the asymptotcally geometrc state probablty. We do not have a formal proof of convergence of ths teratve scheme. In practce, t has converged wthout any dampng n the teraton for all the cases nvestgated. The next secton s devoted to a formal proof that our recurrent algorthm s numercally stable. 4. Stablty of recurrent soluton Our goal n ths secton s to show that our recurrent soluton descrbed n Secton 2 s computatonally stable. For our purposes here, we fnd t convenent to rewrte the recurrence as p( n)λ + µ = p( 1 n)µ 1 ˆq 1 + λδ,1 + p( n 1)u, (28) where we use the followng notaton { 1, = 1 p( 0) = p(0 n) = 0 and δ 0, > 1,,1 = As descrbed n Secton 2, the soluton can be expressed as p( n) = b λ + c u wth the coeffcents b and c gven by }. { 1, = 1 0, 1. b λ + µ = b 1 µ 1 ˆq 1 + δ,1 (30) c λ + µ = c 1 µ 1 ˆq 1 + p( n 1). (31) By conventon, we let b 0 = c 0 = 0. u s gven by (17),.e. / u = 1 λ b c. =1 =1 We assume that the Coxan servce tme dstrbuton has ndeed k stages, so that µ > 0, = 1,..., k, 0 < ˆq 1 for < k and ˆq k = 0. We start by showng that our recurrence for p( n) produces postve values. (24) (29)

7 372 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Lemma 1. If p(1 n 1) > 0 and p( n 1) 0 for > 1, then we have (1) b > 0 and c > 0 for 1, (2) u > 0, and (3) p( n) > 0 for 1. Proof. (1) Follows drectly from (30) and (31). (2) Summng (30) over all values of and usng the fact that ˆq k = 0, we have b λ + µ = =1 b µ ˆq + 1. =1 Rearrangng the terms and usng the fact that q = 1 ˆq, we get 1 λ b = =1 b µ q > 0, and hence u > 0. =1 (3) Follows drectly from the results of (1) and (2). Lemma 1 tells us that p( n) > 0 for 1 and n 1. We now consder a set of perturbed condtonal probabltes (the perturbaton correspondng to floatng pont round-off errors) p( n 1) = p( n 1) + (n 1). Snce the condtonal probablty s normalzed at each step of the recurrence, the perturbatons (n 1) must satsfy k =1 (n 1) = 0. We also assume that the perturbatons are small so that we have p( n 1) > 0. (30) shows that the perturbaton does not affect b. Let c = c + c (n 1) be the soluton of (31) when p( n 1) s replaced by p( n 1). (n 1) and c are related by c λ + µ = c 1 µ 1 ˆq 1 + (n 1). (32) In Lemma 2 we bound relatve perturbatons n p (n 1) n terms of relatve perturbatons n c. Lemma 2. (n 1) mn c c c c mn Proof. See Appendx. and c satsfy (n 1) p( n 1) (n 1) p( n 1). Let p( n) = p( n) + be the condtonal probabltes at n correspondng to p( n 1). p( n) s expressed n (29) as where p( n) = b λ + c ũ, ũ = 1 λ / b c. =1 The next lemma relates to c. =1

8 Lemma 3. where satsfes uc = β β = =1 Proof. See Appendx. A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) / c c. =1 c c β, We now have the elements to prove the stablty of our recurrent algorthm. Consder a functon measurng the magntude of the relatve error n p( n) g = p( n) mn 1 + mn p( n) p( n). Theorem 1. g satsfes (1) g g(n 1) (2) p( n) g. Proof. (1) k =1 = 0 mples 0 and mn p( n) Usng (29) and the results of Lemma 1, we have p( n) uc Substtutng nto functon g yelds g uc mn 1 + mn uc 0. (33) p( n) and uc. mn From the results of Lemmas 2 and 3 t follows uc mn 1 = uc 1 + β and 1 + mn uc = β 1 + mn β p( n) mn β 1 + mn c c (n 1) p( n 1) uc. c c mn c c (n 1) p( n 1) mn > 0. (n 1) p( n 1)

9 374 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Fg. 2. Speed of convergence for Erlang-k servce dstrbuton. Combnng these results we get g g(n 1). (2) Usng (33), we conclude that g p( n) mn Fg. 3. Lmtng servce rate for Erlang-k servce dstrbuton. p( n) p( n). Theorem 1 demonstrates that our recurrent algorthm s numercally stable. In the next secton we present numercal results that llustrate the computatonal behavor of our method. 5. Numercal results We start by llustratng the behavor of our method n the case of nfnte state-ndependent queues. Unless specfed otherwse, the convergence crteron used n our results for nfnte queues was 1 u/u(n 1) < ε for all n 1 wth ε = In all examples presented n ths secton, the mean servce tme was kept at 1. Fg. 2 shows the number of recurrence steps before convergence s reached when the servce tme s an Erlang dstrbuton of order k for k = 2, 4, 8, 16, 32, and 64. More specfcally, we take µ 1 = = µ k = k and ˆq 1 = = ˆq k 1 = 1. The correspondng coeffcent of varaton of the Erlang dstrbuton s 1/ k,.e., 0.71, 0.5, 0.35, 0.25, 0.18, and n our example. We consder three values of server utlzaton: 0.25, 0.5, and 0.999, the latter value corresponds to a system that s barely ergodc. In Fg. 3, we present the correspondng value of the servce rate at convergence,.e., the lmtng value ũ (n the asymptotcally geometrc state probablty dstrbuton p). Fgures are labeled wth the coeffcent of varaton of the servce tme dstrbuton. We observe that, n ths example, convergence occurs wthn a few tens of recurrence steps. Ths seems to be the case for many systems. Interestngly, the value of the lmtng servce rate ũ s generally very dfferent from the nverse of the expected servce tme, and only approaches the latter near server saturaton.

10 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Fg. 4. Example of slower convergence. Fg. 5. Lmtng servce rate for slower convergence case. Although convergence to the lmtng servce rate value tends to occur rapdly for many systems, there are cases when consderably more recurrence steps are needed to acheve the strngent convergence level we used. Fg. 4 shows the number of recurrence steps n one of the worst cases we have encountered. The Coxan servce tme dstrbuton used n ths example corresponds to the followng parameters: k = 9, µ 1 = 2, µ 2 = µ k = 2(k 1) ˆq 1, ˆq 2 = = ˆq k 1 = 1, and the values of ˆq 1 are 0.1, 0.01 and The resultng coeffcent of varaton of the servce tme dstrbuton s 1.68, 5.30, and 16.77, respectvely, and we use t to label the results. In Fg. 5, we have represented the correspondng lmtng servce rates. We observe that the number of teraton steps needed to satsfy the specfed convergence crteron reaches 2500 n ths case. Note that although the number of recurrence steps n ths example ncreases wth the coeffcent of varaton of the servce tme dstrbuton, other hgh varablty dstrbutons exhbt much faster convergence. Fgs. 6 and 7 show the results for a two-stage Coxan wth µ 1 = 2 and the followng sets of parameter values for the second stage and the probablty that stage 2 wll follow the completon of stage 1: set 1: µ 2 = 0.3, ˆq 1 = 0.15; set 2: µ 2 = 0.1, ˆq 1 = 0.05; set 3: µ 2 = 0.01, ˆq 1 = 0.05; set 4: µ 2 = 0.001, ˆq 1 = The resultng coeffcent of varaton s 1.83, 3.16, 10, and 31.62, respectvely. Here we observe that the number of recurrence steps remans below one hundred even for the hghest varablty case and very close to server saturaton. As before, the lmtng servce rate s qute dfferent from the nverse of the mean servce tme (1, n our case), except close to saturaton. Interestngly, for the hgh varablty dstrbuton

11 376 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Fg. 6. Speed of convergence for hgh varablty dstrbuton. Fg. 7. Lmtng servce rate for hgh varablty dstrbuton. Fg. 8. Convergence speed versus convergence strngency. the lmtng servce rate value ũ s lower than the nverse of the mean servce tme (unlke for the Erlang servce dstrbuton). Whle the value of ε n the convergence crteron used n our examples yelds accurate results both for the expected number n the system and for selected state probabltes, t may seem rather strngent. In our next example, we study the nfluence of the convergence strngency on the number of recurrence steps. The results shown n Fg. 8 have been obtaned for a servce tme dstrbuton smlar to that used n Fgs. 4 and 5, except that the total number of stages n the Coxan dstrbuton s 7. The coeffcent of varaton of the servce tme dstrbuton s The results n Fg. 8 are labeled by the server utlzaton.

12 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) Fg. 9. Expected number of customers n fnte source system. Fg. 10. Probablty server s dle n fnte source system. We notce that, at hgher server utlzaton levels, the convergence strngency may have an mportant effect on the number of recurrence steps requred. For the convergence crteron used n these examples, 1 u/u(n 1) < ε for all n 1, t has been our experence that ε = s suffcent to acheve accurate results for most systems wth server utlzaton of up to For server utlzatons not exceedng 0.7, ε = tends to be suffcent. Note that for many systems the number of recurrence steps to convergence s qute low even wth more strngent values for ε, and the soluton tme n all cases s neglgble. The computatonal effort at each recurrence step s mnmal. Note also that the storage requrements of our approach are very low. If we embed the recurrence steps nsde the computaton of state probabltes p for ncreasng values of n, there s no need to store more than the last set of the condtonal probabltes p( n) and the last two values of the servce rate u. It s also suffcent to store only the values of p (and any derved quanttes) one s nterested n. Addtonally, we note that, even n the cases where a larger number of recurrence steps s needed, we have not encountered numercal problems. Snce we know the lmtng servce rate ũ from the soluton of Eqs. (24) and (25), we can double check for any loss of accuracy when the recurrent computaton of u ends due to convergence. As a last example, we consder a system wth state-dependent arrvals where λ = (N n)α. Ths form of the arrval rate corresponds to a set of N customers each generatng a request wth rate α. The results shown n Fgs. 9 and 10 pertan to a system wth N = 100 and the nne-stage hgh varablty servce tme dstrbuton used n Fgs. 4 and 5. In ths fnte-source example, we consder more values for ˆq 1 : 0.5, 0.1, 0.05, 0.01, 0.005, and The resultng coeffcent of varaton for the servce tme dstrbuton ranges from 0.75 to 16.77, and s used as a label n Fgs. 8 and 9. Fg. 9 dsplays the expected number of customers n the system as a functon of the mum arrval rate Nα. Fg. 10 shows the correspondng probablty that the server s dle n such a system versus the same mum arrval rate. It s nterestng to note the effect of servce tme varablty on the expected number of customers n the system, as well as on the probablty that the system s dle. As the coeffcent of varaton of the servce tme dstrbuton ncreases so does the probablty that the server s dle. Ths effect becomes more vsble as the mum arrval rate

13 378 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) ncreases. The expected number of customers ncreases wth servce tme varablty but the relatve ncrease wth servce tme varablty becomes smaller as the arrval rate ncreases. Overall, n all the examples we studed, we have found the proposed method to be numercally well behaved, and generally very fast. Addtonally, our method s qute easy to mplement n a language lke C, C++, Java or FORTRAN, wth mnmal memory space requrements. 6. Concluson We have proposed an approach, beleved to be novel, to the soluton of M/G/1-lke queues n whch the servce tme dstrbuton s represented by a Coxan seres of memoryless stages. Our method s based on elementary condtonal probabltes, and provdes a smple, computatonally effcent and stable recurrence that allows us to obtan statedependent servce rates, and, hence, the steady-state queue length dstrbuton. Unlke exstng methods, the proposed approach does not rely on the use of matrx-geometrc technques or stochastc complementaton. It can handle generalzed queues n whch both the arrval rates and the parameters of the servce tme dstrbuton, ncludng the probabltes of movng from stage to stage, may be state dependent. Our method can be appled to both fnte and nfnte M/G/1-lke queues. We beleve that our method combnes conceptual smplcty, ease of mplementaton and computatonal effcency. In the case of an nfnte, state-ndependent queue, the explct product form of the queue length dstrbuton n our method allows us to show that t s asymptotcally geometrc. Addtonally, we have gven a smple set of equatons that defnes the lmtng servce rate for ths geometrc dstrbuton, and we have presented a smple fxed-pont teraton to solve t. We have provded a proof of the numercal stablty of our algorthm n the general case, and a proof of ts convergence to the unque non-negatve fxed pont n the case of a state-ndependent nfnte queue. We have ncluded several examples to llustrate the performance of our method n the case of an nfnte M/G/1-lke queue. We have consdered both low and hgh servce tme varablty, and server utlzatons rangng from low to very close to saturaton. Our results for nfnte queues suggest that, n general, only a moderate number of recurrence steps s needed to reach the lmtng servce rate wth hgh accuracy. In many cases, ust a few tens of recurrence steps may be suffcent. We have also ncluded an example to llustrate the results obtaned wth our method n the case of a fnte number of customers and state-dependent arrvals. The proposed method s n general fast, very thrfty n terms of memory space requrements, conceptually smple, and easy to mplement. It s stable even for systems wth server utlzaton very close to saturaton. Its smplcty and numerc stablty should make t attractve to performance analysts. An extenson of ths approach to M/G/1-lke systems wth servce nterruptons s under study. Acknowledgments The authors wsh to thank the anonymous referees for ther remarks and suggestons whch helped mprove ths paper. Appendx A A.1. Proof of lemmas c c Proof of Lemma 2. Suppose c m c m λ + µ m c m s attaned at = m. At = m, we wrte (32) as c m 1 c m 1 c m 1 µ m 1 ˆq m 1 = m (n 1) p(m n 1) whch leads to c m (n 1) m {c m λ + µ m c m 1 µ m 1 ˆq m 1 } p(m n 1). c m p(m n 1) p(m n 1)

14 On the other hand, at = m, (31) becomes A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) c m λ + µ m c m 1 µ m 1 ˆq m 1 = p(m m 1). Combnng these two results, we obtan c c = c m c m (n 1) m p(m n 1) Usng an analogous approach, we can show that mn c c mn (n 1) p( n 1). (n 1) p( n 1). Proof of Lemma 3. u = ũ u 1 λ k b = = =1 1 λ k b =1 c + c c =1 =1 ( ) ( ) 1 λ k b =1 c =1 β ( ) ( ) = u c c + k c 1 + β =1 =1 =1 = p( n) p( n) = u + u c + c uc = u c + uc + c = u c β 1 + β c + c = u c 1 + β β c. Proof of Lemma 4. Usng (29) and the fact that p( n) λb p( n) 1 λ mn b = α, we obtan mn ( p( n) λb p( n) = p( n) ( p( n) λ b p( n) = mn p( n) Substtutng nto functon g yelds g α uc mn 1 + mn uc uc. ) uc ) uc α α mn uc uc. Then g αg(n 1) follows from the proof of Theorem 1.

15 380 A. Brandwan, H. Wang / Performance Evaluaton 65 (2008) A.2. Convergence to fxed pont for nfnte state-ndependent queue Consder now the nfnte state-ndependent queue of Secton 3. For an nfnte state-ndependent queue, all parameters are ndependent of n. As a results, the coeffcents b defned n (30) are also ndependent of n. Let α = 1 λ mn b. From Lemma 1 we get 0 < α < 1, and we have Lemma 4. For an nfnte state-ndependent queue, g satsfes g αg(n 1). Proof of Lemma 4. See Appendx A.1. For notatonal convenence, we let p = (p(1 n), p(2 n),..., p(k n)) T. We have Theorem 2. The sequence { p, n = 1, 2,..., } converges to a unque fxed pont. Proof of Theorem 2. The key to the proof s to vew p(n + 1) as p perturbed: and p( n + 1) = p( n) +. Usng the results of Theorem 1 and Lemma 4, we have p(n + 1) p = p(n + m) p(n) N+m 1 n=n p( n) g αn g(1) p(n + 1) p ( N+m 1 n=n α n ) g(1) α N g(1) 1 α. Thus, { p, n = 1, 2,..., } s a Cauchy sequence wth respect to the nfnty norm, and converges wth respect to ths norm to a lmt p( ). Snce p( ) s the lmt of both p and p(n + 1), t must be a fxed pont. Now we show that our recurrent algorthm has only one non-negatve fxed pont. Suppose p = (p(1), p(2),..., p(k)) T s a fxed pont of our recurrent algorthm. Substtutng nto (28), we have p()(λ + µ ) = p( 1)µ 1 ˆq 1 + λδ,1 + p()u, where u = Let us treat u as a parameter and wrte p() as p() = p( 1)µ 1 ˆq 1 + λδ,1 /(λ + µ u). p( )µ q. (34) To ensure that p() s non-negatve, we must have u < mn (λ + µ ). It s clear that for u < mn (λ + µ ), p() s a strctly ncreasng functon of u. When u = 0, summng (34) over all values of and dvdng by λ yelds k=1 p() = 1 λ 1 k=1 p()µ q < 1. As u mn (λ + µ ), k =1 p() + > 1. Snce k =1 p() s a contnuous and strctly ncreasng functon of u, there s a unque value of u such that k =1 p() = 1 s satsfed. Ths means that the non-lnear equaton (34) has a unque non-negatve soluton. References 1 N. Akar, K. Sohraby, An nvarant subspace approach n M/G/1 and G/M/1 type Markov chans, Commun. Statst. Stochastc Models 13 (1997) A.O. Allen, Probablty, Statstcs, and Queung Theory, 2nd edton, Academc Press, N.G. Bean, P.K. Pollett, P.G. Taylor, Quasstatonary dstrbutons for level-dependent quas-brth-and-death processes, Commun. Statst. Stochastc Models 16 (2000) N.G. Bean, J.-M. L, P.G. Taylor, Caudal characterstcs of QBDs wth decomposable phase spaces, n: G. Latouche, P.G. Taylor (Eds.), Advances n Algorthmc Methods for Stochastc Models, Notable Publcatons, NJ, 2000, pp D.A. Bn, G. Latouche, B. Men, Numercal Methods for Structured Markov Chans, Oxford Unversty Press, =1

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